Machinery's Handbook, 31st Edition
Equation Solving
31
Equation Solving An equation is a statement of equality between two expressions, such as one monomial set equal to another, like 5 x = 105. The unknown, or variable, is frequently designated by x . Other unknowns (if any) are designated by letters also usually selected from the end of the alphabet: y , z , u , t , etc. Equations, like expressions, have a degree. A first-degree equation is one in which the variable is raised to the first power, as in 3 x = 9. A second-degree equation, also called a quadratic equation, is one in which the highest power of the variable is two; for example, x 2 + 3 x = 10. Solving a first-degree equation requires isolating the unknown. In the example below, x is the unknown variable. To get x alone (to isolate x ), constants are combined on one side of the equation and variable terms are combined on the other. The steps are: Given: 10 x – 14 = 8 – 2 x Add 2 x to both sides: 10 x – 14 + 2 x = 8 – 2 x + 2 x 12 x – 14 = 8 Add 14 to both sides: 12 x – 14 + 14 = 8 + 14 12 x = 22 Divide by 12 (multiply by 1/12): 12 x /12 = 22/12 Simplify: x = 22/12 = 11/6 Any answer can be checked by substituting it into the original equation to see that it satisfies it. Solving a System of Linear Equations.— More involved than solving a single-variable equation is the process of solving a system of linear equations. A simple linear system rep- resents two lines in the plane that behave in one of three ways: they intersect at one point, in which case they have a unique solution , ( x, y ); they intersect everywhere—that is, they are collinear —so all points ( x, y ) of one line satisfy the other; or they are parallel and thus intersect nowhere, hence, there is no solution. The methods for solving a system of linear equations are substitution and elimination , as shown next. Substitution: In this method, one of the variables is expressed in terms of the other variable by isolating it. This expression is then substituted into the second equation, con- verting it to a single-variable equation. It is solved for this variable, and the solution is substituted back into the either of the original two equations to find the value of the other variable. Example (Unique Solution): Find the ordered pair ( x, y ) that satisfies the system of equations: 2 7 2 4 x y x y + = − = − First, solve either equation for one variable in terms of the other. Say, solve the second equation for x : x = 2 y – 4 Then, substitute this expression for x into the first equation and solve it for y : 22 4 7 y y y y y y ( ) + = → − + = → = → = 4 8 7 5 15 3 – Finally, substitute y = 3 into the second equation: x = 2(3) – 4 = 6 – 4 = 2. The solution (that is, the point at which the lines intersect) is (2, 3). Example (Infinite Solutions): Find the ordered pair ( x, y ) that satisfies the system of equations: 7 3 14 2 6 x y x y − = − =
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